Due to the increasing complexity of electronic systems—caused by, e.g., higher integration density, increased functionality, etc.—a structured, for instance, a “top down”, “bottom up,” or similar conventional approach to design has become necessary.
In a top down approach for instance, the design of a particular electronic system is started at a relatively high level of abstraction, whereafter the respective design is progressively refined at increasingly lower levels of abstraction (e.g., starting from a “system module level” down to a “circuit level”, etc.).
Thereby, at each level of abstraction, appropriate tests and/or simulations are carried out; in case of faults the design result must be modified, and/or the corresponding design step repeated, or the design started anew at a higher level.
This design approach ensures that, despite the increased system complexity, the designed system operates in a fault-free fashion.
The test/simulation of so-called multi-tone systems is of particular importance.
In such systems, the multiple occurring input frequencies (“tones”) (i.e., the frequencies of the input signals frf1, frf2, . . . , and the frequency of a local oscillator flo, etc.) are in general incommensurable; hence, no common base frequency exists for such a system.
A steady state x(t) (vector containing all unknown variables) of a multi-tone system with incommensurable frequencies is quasi-periodic and may be represented as a multi-dimensional Fourier series (for sake of simplicity here given for two frequencies (“tones”) f1 and f2, only)
      x    ⁡          (      t      )        =            ∑              -        ∞            ∞        ⁢                  ∑                  -          ∞                ∞            ⁢                        c                                    i              1                        ⁢                          i              2                                      ⁢                  ⅇ                      j            ⁢                                                  ⁢                          i              1                        ⁢            2            ⁢            π            ⁢                                                  ⁢                          f              1                        ⁢            t                          ⁢                  ⅇ                      j            ⁢                                                  ⁢                          i              2                        ⁢            2            ⁢            π            ⁢                                                  ⁢                          f              2                        ⁢            t                              
In the state of the art, methods to compute such quasi-periodic steady states are well-known, e.g., from V. Rizzoli, C. Cecchetti, A. Lipparini, F. Mastri, general-purpose harmonic balance analysis of nonlinear microwave circuits under multitone excitation, IEEE transactions on microwave theory and techniques, vol. 36, No. 12, December 1988, pages 1650-1660, the contents of which is incorporated herein by reference.
According to such methods, the unknown vector of multi-dimensional Fourier coefficients (ci1i2) might be computed concentrating on combinations of frequencies (“mixing products”) of interest only, and, e.g., not taking into account the further combinations of frequencies.
For example, from the set of possible linear combinations of the n tones of the system:
              ∑              j        =        1            n        ⁢                  i        j            ⁢              f        j              (ij integer, mixing order≦ij≦mixing order, j=1, . . . , n)for the above computation, only frequencies up to a certain mixing order might be used. For example, only those frequencies as selected according to, e.g., one of the “diamond rule” and the “box rule”:
      Diamond    ⁢                  ⁢    rule    ⁢          :                          ∑                  j          =          1                n            ⁢                                i          j                              ≤          mixing      ⁢              -            ⁢      order        or                    ∑                  j          =          1                n            ⁢              i        j              =    0        Box    ⁢                  ⁢    rule    ⁢          :                                i        j                    ≤          mixing      ⁢              -            ⁢      order      
However, since the above system equations for the multi-dimensional Fourier series in general are nonlinear, the solution can only be obtained by applying an iterative solver, e.g., “Newton's method”. An iterative solver such as “Newton's method” requires an initial guess of the solution (i.e., for the unknown Fourier coefficients).
A bad guess might lead to a poor and time-consuming convergence, or even to a failure of the respective iterative method for computing the unknown Fourier coefficients.
Even in case of convergence, a bad guess might lead to false solutions due to an incorrect initialization of circuit parts that comprise memory, e.g., frequency dividers.